Remarks on extremal and maximum determinant matrices with moduli of real entries less than or equal to 1
This note discusses quasi-orthogonal matrices which were first highlighted by J. J. Sylvester and later by V. Belevitch, who showed that three level matrices mapped to lossless telephone connections. The goal of this note is to develop a theory of such matrices based on preliminary research results. Methods: Extreme solutions (using the determinant) have been established by minimization of the maximum of the absolute values of the elements of the matrices followed by their subsequent classification. Results: We give the definitions of Balonin-Mironovsky (BM), Balonin-Sergeev (BSM) and Cretan matrices (CM), illustrations for some elementary and some interesting cases, and reveal some new properties of weighing matrices (Balonin-Seberry conjecture). We restrict our attention in this remark to the properties of Cretan matrices depending on their order.